1202.1520 (Roger E. Behrend et al.)

The doubly-refined enumeration of alternating sign matrices and
descending plane partitions    [PDF]

Roger E. Behrend, Philippe Di Francesco, Paul Zinn-Justin

It was shown recently [R. Behrend, P. Di Francesco and P. Zinn-Justin, On the
weighted enumeration of alternating sign matrices and descending plane
partitions, J. Combin. Theory Ser. A 119 (2012), 331-363] that, for any n,
there is equality between the distributions of certain triplets of statistics
on nxn alternating sign matrices (ASMs) and descending plane partitions (DPPs)
with each part at most n. The statistics for an ASM A are the number of
generalized inversions in A, the number of -1's in A and the number of 0's to
the left of the 1 in the first row of A, and the respective statistics for a
DPP D are the number of nonspecial parts in D, the number of special parts in D
and the number of n's in D. Here, the result is generalized to include a fourth
statistic for each type of object, where this is the number of 0's to the right
of the 1 in the last row of an ASM, and the number of (n-1)'s plus the number
of rows of length n-1 in a DPP. This generalization is proved using the known
equality of the three-statistic generating functions, together with relations
which express each four-statistic generating function in terms of its
three-statistic counterpart. These relations are obtained by applying the
Desnanot-Jacobi identity to determinantal expressions for the generating
functions, where the determinants arise from standard methods involving the
six-vertex model with domain-wall boundary conditions for ASMs, and
nonintersecting lattice paths for DPPs.

View original: http://arxiv.org/abs/1202.1520